How many 2 x 2 matrices are there with entries from the set {0, 1, ..., k} in which there are no zero rows and no zero columns.

My solution

No. of matrices in which there are zero rows$\displaystyle =2(k+1)^2-1$ (Assume that the first row is a zero row. There are $\displaystyle (k+1)(k+1)=(k+1)^2$ ways of choosing the elements of the second row. Since the second row could as well be the zero row, we get $\displaystyle 2(k+1)^2$ ways. Since we have counted the zero matrix twice, we must subtract 1.)

No. of matrices in which there are zero columns$\displaystyle =2(k+1)^2-1$

No. of matrices in which there are both zero rows and zero columns$\displaystyle =4k+1$ ($\displaystyle 4k$ matrices each with one non-zero element and 1 zero matrix.)

No. of matrices in which there are either zero rows or zero columns$\displaystyle =2[2(k+1)^2-1]-(4k+1)=4k^2+4k+1$

Total no. of matrices$\displaystyle =(k+1)^4$

No. of matrices in which there are no zero rows and no zero columns$\displaystyle =(k+1)^4-(4k^2+4k+1)=k^4+4k^3+2k^2$

Could someone please check if my solution is correct? Thanks.